Fourier Transforms Flashcards
sinc(x) =
sinx/x
proof of fourier transform
rk = r2π/T = ωr
Δω = 2π/T
1/T = Δω/2π
substitute into complex fourier series.
Let T -> inf
Σ Δω -> inf ∫ -inf
Properties of the Fourier Transform
FT twice almost the original function
Linearity of FT
Scaling
Translation
Differentiation
Integration
Scaling of Fourier Transform
a > 0 t/a -> t’ dt = adt t = at’
= a/√2π (inf ∫ -inf) dt’ f(at’) exp[-iωt]
f(ω/a) = 1/√2π (inf ∫ -inf) dt f(at) exp[-iωt]
Translation of Fourier Transform
t+a = t’ dt = dt’
1/√2π (inf ∫ -inf) dt f(t’) exp[-iω(t’-a)]
= exp[iωa] 1/√2π (inf ∫ -inf) dt f(t) exp[-iωt]
apply fourier transform to both sides.
F[f(x-a)] = 1/√2π (inf ∫ -inf) dt f(t’) exp[-iω(t’+a)]
Solving Differential Equations using Fourier Transforms
1) Differentiate Eq
2) FT of both sides
3) Exploit Linearity
4) Exploit Derivative
5) Rearrange for f(ω) and take the inverse
FT of a gaussian is
another gaussian
taking the fourier of sine and cosine
convert exponential to cos(ωt) - isin(ωt)
Scaling property of the delta dirac function
(inf ∫ -inf) f(t) δ(at) dt = (inf ∫ -inf) f(t’/a) δ(t’) dt’/a
= 1/a f(0)
= 1/a (inf ∫ -inf) f(t) δ(t) dt
= (inf ∫ -inf) f(t) δ(t)/a dt
identify δ(at) = δ(t)/a
then do for opposite limit sign
giving δ(at) = - δ(t)/a
for higher derivatives - delta dirac
(inf ∫ -inf) dt f(t) δ^(n) t = (-1)^(n) f^(n) (0)
Fourier transform of the delta function
f(t) = 1/2π (inf ∫ -inf) du (inf ∫ -inf) dv f(v) exp[iω(t-u)]
(inf ∫ -inf) dv f(v) δ(u-t) = f(t)
δ(t) = 1/2π (inf ∫ -inf) dω exp[iωt]
δ(t) = 1/√2π (inf ∫ -inf) 1/√2π exp[iωt] dω
δ(tilda)(ω) = 1/√2π
For the fourier transform x -> ∞
The FT exists if the integral is finite for all times t
therefore f(x) -> 0 as x -> ∞
Fourier domain PDE
take fourier transform of both sides
the PDE is now an ODE
take FT of initial conditions for general solution
convolution applications
in photography
the measurements are given by the convolution of the signal and resolution function
